Simplify the following expression and state the condition under which the simplification is valid: $n = \dfrac{t^2 - 4t - 45}{t^2 + t - 20}$
Solution: First factor the expressions in the numerator and denominator. $ \dfrac{t^2 - 4t - 45}{t^2 + t - 20} = \dfrac{(t - 9)(t + 5)}{(t - 4)(t + 5)} $ Notice that the term $(t + 5)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(t + 5)$ gives: $n = \dfrac{t - 9}{t - 4}$ Since we divided by $(t + 5)$, $t \neq -5$. $n = \dfrac{t - 9}{t - 4}; \space t \neq -5$